MathDB

p1. For what values of

a

(5 a + 1)(3a + 2)

divisible by

15

?
p2. Of the four-digit positive integers that are multiple of

9

, how many are there that have all their digits different from zero and different from each other?
p3. A square with side

5

is divided into

25

unit squares by lines parallel to the sides. Let A be the set of the

16

interior points, which are vertices of the unit squares, but which are not on the sides of the initial square. What is the largest number of points in

A

that can be chosen so that any three of them not vertices of a right isosceles triangle?
p4. In a parallelogram

ABCD

BC = 2AB

. The bisector of angle

B

intersects the extension of segment

CD

Q

and the bisector of angle

A

intersects segment

BD

M

and segment

BC

P

. If

PQ = 6

, find the length of the segment

MB

.
p5. Find the number of ways to write nonnegative integers in each box on a board of dimension

n \times n

, so that the sum of the numbers in each row and each column is equal to

3

and in each row and in each column there are only one or two numbers different from zero.

Problem Statement